Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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2
votes
1answer
71 views

Find all the $2 \times 2$ matrices $A$ satisfying $A^2 = 0$

Find all the $2 \times 2$ matrices $A$ satisfying $A^2 = 0$. So far I have this: But I don't know to proceed.
1
vote
3answers
43 views

Assume that A and B are square matrices, so that $AB = B^2A$. Prove $(AB)^2=B^6A^2$. [on hold]

I have no idea how to solve this or if I miss any properties,I will appreciate any kind of help and explanation. Excuse me for my broken english, is not my first language
9
votes
2answers
58 views

$\operatorname{rank} A = \operatorname{rank} A^2$ if and only if $\lim_{\lambda \to 0} (A+\lambda I)^{-1}A$ exists

Let $A$ be any complex $n\times n $ matrix. Prove that $\operatorname{rank} A = \operatorname{rank} A^2$ if and only if $\lim_{\lambda \to 0} (A+\lambda I)^{-1}A$ exists. I am stuck on this problem, ...
1
vote
0answers
44 views

Is my understanding of matrices correct?

Let $V = R^2$ be our vector space with the unit base vectors $J(1, 0), K(0, 1)$. We have the linear map, $$T: V \to V$$ $$T(v) = v'$$ We can rewrite $\forall v \in V$ as a linear combination of the ...
0
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1answer
16 views

Direct proof for angular velocity from direction cosine matrix

I am trying to work through the math of what should be a relatively simple proof of a direct definition of the angular velocity matrix starting from the direction cosine matrix. The reference for ...
3
votes
1answer
47 views

Which vectors are stretched the most in a transform?

I thought that it would be obvious that in a transform the eigenvectors are the ones that are stretched the most as those are the directions in which the matrix acts. But according to this short video ...
-1
votes
1answer
22 views

Rank of product of a matrix and its Moore-Penrose inverse [on hold]

Given a m x n matrix X with rank r: Does the product of X and its MP inverse always have rank r? If yes, why? Thanks
0
votes
1answer
37 views

Which one is correct between “$(Ax=0)$ $\land$ $(x\neq 0)$ $\iff det(A)=0$ ” and “$(Ax=0)$ $\land$ $(x\neq 0)$ $\implies det(A)=0$”?

As referred to Why non-trivial solution only if determinant is zero, that says "$(A−\lambda I)x=0$ has a nontrivial solution (a solution where $x\neq 0$) if and only if $\det(A−\lambda I)=0$ " which ...
1
vote
1answer
61 views

Question regarding the similarity of an invertible matrix with its inverse .

Find the set $S$ of all possible $n×n$ invertible matrices $M$ such that $M$ is similar to $M^{-1}$ . My approach Actually, I was thinking about this problem when I came across a theorem stating ...
1
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0answers
25 views

Are the zero eigenvalues of a Laplacian matrix semi-simple?

It is known that the Laplacian matrix $\mathcal{L}$ for a directed weighted graph has at least one zero eigenvalue. If it has more than one zero eigenvalue, will there be non-trivial Jordan blocks ...
3
votes
1answer
33 views

Trace of matrix $A^{\ast}A$

Given a $n \times n$ matrix $A$ with complex entries. And $A^{\ast}$ represents the conjugate transpose of $A$.Then If $\left | tr{\left ( A^{\ast}A \right )}\right | <n^2$, then $\left |a_{...
2
votes
1answer
23 views

Determine whether elements of a set are bounded above, below, or to the side of another in 3 or more directions

I'm having a difficult time expressing an algorithm mathematically. Assuming a 2D matrix of $x$ and $y$ elements, I have a set of pairs for elements that have been found and definitely exist ($A$), ...
0
votes
1answer
19 views

Positive-Definite Matrix Question

I want to prove that the matrix is positive definite using the fact that: If $A$ is symmetric and $\langle x, Ax \rangle$ > $0$ for a nonzero vector $x$ then $A$ is positive. So I have the ...
0
votes
1answer
70 views

Do $A$ and $A^T A$ share an eigenvector?

I have been learning about singular value decomposition from http://www.ams.org/publicoutreach/feature-column/fcarc-svd and they say that orthongoal vectors in the domain are mapped to orthogonal ...
-3
votes
1answer
39 views

(A−λI)x=0 and x≠0 iff det(A−λI)=0: Why [[1,1],[1,1]][[2],[3]] = [[5],[5]] ≠ 0 when det([[1,1],[1,1]]) = 0?

As refered to Why non-trivial solution only if determinant is zero, I wonder why \begin{gather} \begin{bmatrix} 1 & 1 \\1 & 1 \end{bmatrix} \begin{bmatrix} 2 \\3 \end{bmatrix} = \begin{...
1
vote
2answers
32 views

Given a Hermitian matrix $A$, prove that $(A-iI)$ is nonsingular

The exercise is to prove that, given $A$ a Hermitian matrix, then $(A-iI)$ is nonsingular. I tried to think about what it meant to be nonsingular, like $(A-iI)X=0$ have not only the trivial solution, ...
0
votes
1answer
38 views

Is it always possible to swap columns of a matrix by a left hand side multiplication?

I was thinking about swapping the columns of the matrix. It is well known that if you want to swap 2 columns of a matrix, you do a right hand side multiplication with a permutation matrix $T_{ij}$, ...
0
votes
0answers
9 views

Parameterizing rotation matrix

A general rotation matrix in terms of Euler angles is given by $$ \mathcal{R}=R_{\hat z}(\alpha)R_{\hat y}(\beta) R_{\hat z}(\gamma). $$ Working out the matrix multiplication we obtain the known ...
2
votes
0answers
21 views

trace of the product of two projection matrices [on hold]

If $P_{i}=\frac{\alpha_{i}\otimes\beta_{i}}{(\alpha_{i},\;\beta_{i})}=\frac{\alpha_{i}\beta_{i}^{T}}{\alpha_{i}^{T}\beta_{i}}$, where $P_{i}$ are rank-1 projection matrices and $(\alpha_{i},\;\beta_{i}...
0
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0answers
19 views

Controllability Gramian via integration by parts

I'm trying to program something like this article. On page 3, there's the following equation (eq. 6) that should be expressible in closed-form: $$\int_0^te^{(A(t-t'))} M e^{(A^T(t-t'))}dt'$$ where $...
0
votes
3answers
32 views

Finding n-th power of a 2*2 matrix with 2 identical eigen values

If$$ A = \begin{pmatrix} 3 & -4 \\ 1 & -1 \\ \end{pmatrix} $$ prove that $$A^k = \begin{pmatrix} 1+2k & -4k \\ k & 1-2k \\ \end{pmatrix}$$ Now the first ...
-1
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0answers
37 views

How to multiply vectors of a same matrix?

Let's, for example, consider the $3\times 6$ block matrix $$ \begin{bmatrix} 2&4&0 &\space &\space8&6&2\\ 5&9&0 &\space &\space 1&5&4 \\ 4&7&...
3
votes
3answers
222 views

Showing that the limit of non-eigenvector goes to infinity

Let $A$ be a $3$ by $3$ real matrix with the triple eigenvalue $1$. Also, further suppose its eigenspace corresponding to $1$ is only of dimension $1$. Thus, we can find a basis of $\mathbb{R}^3$, ...
1
vote
3answers
61 views

For the purposes of DFT, is “the” primitive root of unity $w_n = e^{ 2\pi i / n }$ or $w_n = e^{-2\pi i / n }$?

I'm working on the section of Strang's Linear Algebra and Its Applications 4e that discusses discrete Fourier transforms. To make the exercises easier, I wrote myself a Python script that generates ...
0
votes
0answers
39 views
+50

Gauss-Newton Method not converging for my function

I need to solve the optimization problem $$ \min_{x\in \mathbb{R}^{3}}f(x) $$ where the function $f$ is defined as follows: $$ f(x_{1}, x_{2}, x_{3}) = \frac{1}{2}\left[\left(2x_{1}-x_{2}x_{3}-1 \...
-2
votes
0answers
24 views

Probability density function after adding values to the set and transforming

Suppose I have matrix $\mathbf{N}$ (of size $m$ by $n$) of values that has the PDF $f(X)$. Then, I vertically concatenate $\mathbf{N}$ on top of a zero matrix of size $k$ by $n$, like this: \begin{...
1
vote
0answers
31 views

Method for expressing determinant as product of 2 determinants.Again is there any for matrices?

Is there any specific technique to write a determinant as a product of 2 determinants. I have so long kept it doing manually.At the same time I want to ask can any matrix be expressed as a product of ...
0
votes
1answer
42 views

Linear algebra and matrix kernels

I have a set of vectors $v_i \in \mathbb{R}^n$, $i=1,...,m$ and I want to know if there exists a matrix $A \ne 0$ such that $v_i \in \operatorname{Ker}(A)$ for all $i$. Now my approach is to show ...
2
votes
4answers
63 views

Matrixes of higher order like $M_{\aleph\times \aleph}$ [on hold]

I was wondering if thinking about matrixes of the form $M_{\aleph\times \aleph}$ and assigning properties to them like matrix multiplication makes sense and useful in some way. note: $\aleph$ is the ...
3
votes
2answers
58 views

How to convert a $3\times 5$ matrix to a product of a $3\times 2$ and $2\times 5$ matrix?

I have a matrix $$A = \begin{bmatrix} 1 & 1 & 2 & 2 & 3\\ 1 & 0 & 1 & 2 & 2\\ 0 & 1 & 1 & 0 & 1\\ \end{bmatrix},$$ and I would like to write this ...
5
votes
3answers
66 views

Matrix Multiplication - Undefined Product

I am learning linear algebra for a machine learning class and have a question about matrix multiplication. The product of two matrices is undefined whenever the rows of the first matrix (reading right ...
4
votes
1answer
68 views

How do I derive the following expression for the sum of orthogonal matrices?

In Johansen's book 'Likelihood-based inference in cointegrated vector autoregressive models', in order to get the expression for the Granger's representation theorem he claims that: $$\beta_\bot(\...
2
votes
1answer
44 views

How to show the interpretability of NMF by a small qualitative example on a toy data?

In some paper, such as, Nonnegative Matrix Factorization: A Comprehensive Review, I see the interpretability of Nonnegative matrix factorization (NMF). However, I don't know the means of this. How to ...
0
votes
1answer
37 views

Matrix with respect to other basis

In this example from Axler's Linear Algebra done right, The map $T(x,y,z) = (2x+y,5y+3z,8z)$ has a matrix with respect to the standard basis given by $T(1,0,0) = (2,0,0)$ , $T(0,1,0)=(1,5,0)$ , $T(0,...
0
votes
0answers
40 views

Dimension of solution space of the the matrix equation AX=0

Q.2 : Let $T \in \textsf{M}_{m \times n} (\mathbb R)$. Let $\textsf V$ be a subspace of $\textsf{M}_{n \times p} (\mathbb R)$ defined by $$\textsf V = \{ X \in \textsf{M}_{m \times n} (\mathbb R) : \...
0
votes
0answers
28 views

A companion matrix is similar to its transpose

Let $A$ be an $n\times n$ matrix over $\mathbb{C}$. If $A$ is a Jordan block of size $n$, then it can be easily seen that this matrix is similar to its transpose (See comment by Ted here). Now ...
1
vote
0answers
28 views

Smith Normal form of a matrix

Let $A$ be a matrix over $\mathbb{Z}$ of the following form. $$A= \begin{bmatrix} 1 & 0 & \cdots & 0\\ 0 & a_{22} & \cdots & a_{2n}\\ \vdots & \cdots & \ddots & \...
1
vote
1answer
65 views

Existence of a symmetric matrix $X$ such that $XBX = A$

let $A,B$ be two positive semi-definite $n\times n$-matrices such that $$\mathrm{Range}(B^{1/2}AB^{1/2})=\mathrm{Range}(B)$$ and $$\mathrm{Rank}(A)=\mathrm{Rank}(B)=n-1$$ so is there a real ...
4
votes
2answers
83 views

Is there a geometric interpretation about the euclidean distance between of 2 matrices?

The Euclidean distance between points p and q is the length of the line segment connecting them ($\overline{\mathbf{p}\mathbf{q}}$). $$\begin{aligned}d(\mathbf {p} ,\mathbf {q} )=d(\mathbf {q} ,\...
1
vote
2answers
28 views

Why could this formula “dist(x, y) = sqrt(dot(x, x) - 2 * dot(x, y) + dot(y, y))” compute the euclidean distance of two matrices?

sklearn's doc gives this formula for sklearn.metrics.pairwise.euclidean_distances dist(x, y) = sqrt(dot(x, x) - 2 * dot(x, y) + dot(y, y)) to compute the ...
0
votes
2answers
57 views

Are all matrix groups (Lie groups) known

Just wondering if there are any matrix groups out there waiting to be discovered, or if they are all known?
1
vote
1answer
46 views

Derivative of trace of a product containing an inverse matrix

What's the derivative of $$f(X)=\text{Tr}(YX^{-1})$$ with respect to $X$, where $X$ and $Y$ are square matrices of the same dimension? My first attempt is to apply the chain rule as: Let $h(X)=X^{-1}$...
-1
votes
1answer
34 views

Trace of matrix : cannot establish similarity invariant

The similarity invariance for a trace is: $$ \text{Tr}(B)=\text{Tr}(P^{-1}BP) $$ Suppose the following matrix B $$ B=\pmatrix{ a & b \\ c & d } $$ The trace of $B$, that is $\text{Tr}(B)=a+...
-1
votes
1answer
21 views

$\forall C\in \mathbb{R}^{n\times n}, \ \ \ PAP-CMC^T\geq 0\ \ \ \Longrightarrow\ \ \ A-CMC^T\geq 0$ [on hold]

let M be a positive definite $n\times n$ matrix and A a positive semidefinite $n\times n$ matrix and P is an orthogonal projector of some subspace of $\mathbb{R}^n$ into $\mathbb{R}^n$ so is this ...
2
votes
1answer
38 views

Comparison of two quadratic forms

Let A be a positive semi-definite $n\times n$-matrix and P is an orthogonal projector of some subspace of $\mathbb{R}^n$ into $\mathbb{R}^n$ is it correct that $$\forall x\in \mathbb{R}^n\ \ \ x^...
0
votes
0answers
6 views

Argmin using differentiation with respect to inverse matrix

I'm trying to understand the following step in a calculation: My problems: (1) If we want the argmin with respect to $R$, why are we not differentiating with respect to $R$? I assume differentiating ...
3
votes
3answers
71 views

Usefulness of Why Eigenvectors Corresponding to Distinct Eigenvalues of Symmetric Matrix are Orthogonal

I didn't find a pre-existing question relating to what I'm going to ask, so I apologize if this is a duplicate question for one I hadn't found: Why is the property that eigenvalues corresponding to ...
1
vote
3answers
44 views

Diagonalization of a simple matrix. Arriving at a contradiction

Consider the matrix \begin{equation} \begin{pmatrix}1 & 1 & 1\\ 1 & 1 &1\\ 1 & 1 &1\end{pmatrix}.\end{equation} The rank of this matrix is $1$ which is less than $3$. So all ...
2
votes
0answers
15 views

Upper bound the maximum column sum of a particular stochastic matrix

Let $(x_i)_{i=1}^N$ be a set of vectors in $\mathbb{R}^D$. Define the matrix $W \in \mathbb{R}^{N \times N}$ as: $W_{ij} = \frac{\exp(-||x_i-x_j||^2)}{\sum_k \exp(-||x_i-x_k||^2)}$ i.e. row $i$ of $...
0
votes
0answers
8 views

Plucker line coordinates convention in Multiple View Geometry?

In the book from Hartley and ZisserMan "Multiple View Geometry", plucker coordinates were presented as being some cells of the skew-symmetric matrix that is the wedge product of two homogeneous ...